Brush the bottom and sides of a 9-inch square baking pan with vegetable oil. Using a small, fine-mesh sieve, dust the pan generously with confectioners' sugar, knocking out any excess.
Put 1/2 cup water in the bowl of a stand mixer fitted with the whisk attachment. Sprinkle the gelatin into the bowl and stir briefly to make sure all the gelatin is in contact with water. Let soften while you make the sugar syrup.
In a heavy 3- to 4-quart saucepan, combine the granulated sugar, corn syrup, salt, and 1/2 cup water. Place over moderate heat and bring to a boil, stirring until the sugar is dissolved. Put a candy thermometer into the boiling sugar syrup and continue boiling (the mixture may foam up, so turn the heat down slightly if necessary), without stirring, until the thermometer registers 240°F (soft-ball stage). Remove the saucepan from the heat and let stand briefly until the bubbles dissipate slightly.
With the mixer on low speed, pour the hot sugar syrup into the softened gelatin in a thin stream down the side of the bowl. Gradually increase the mixer speed to high and beat until the marshmallow is very thick and forms a thick ribbon when the whisk is lifted, about 5 minutes. Beat in the vanilla.
Scrape the marshmallow into the prepared pan (it will be very sticky) and use wet fingertips to spread it evenly and smooth the top. Let stand, uncovered at room temperature, until the surface is no longer sticky and you can gently pull the marshmallow away from the sides of the pan with your fingertips, at least 4 hours or overnight.
Dust a cutting board with confectioners' sugar. Use a rubber spatula to pull the sides of the marshmallow from the edge of the pan (use the spatula to loosen the marshmallow from the bottom of the pan if necessary) and invert onto the cutting board. Dust the top with confectioners' sugar. Brush a long thin knife or a chef's knife with vegetable oil and dust with confectioners' sugar to prevent sticking; continue dusting the knife as necessary. Cut lengthwise into 8 strips, then crosswise into eighths, to form a total of 64 squares. (For larger marshmallows, cut lengthwise into 6 strips, then crosswise into sixths, to form a total of 36 squares.) Coat marshmallows, one at a time, in confectioners' sugar, using a pastry brush to brush off any excess.
Marshmallows can be stored, layered between sheets of wax paper or parchment in an airtight container in a dry place at cool room temperature, for 1 month.